Working with the context of a theory proposed recently by Fried et al. (2001), we consider a one-dimensional problem involving granular mixture of K > 2 discrete sizes bounded below by an impermeable base, above by an evolving free surface, and subject to gravity. We demonstrate the existence of a solution in which the medium segregates by particle size. For a mixture of small and large particles (K = 2), we use methods of Smoller (1994) to show that the segregated solution is unique. Further, for a mixture of small, medium, and large particles (K = 3), we use LeVeque's (1994) CLAWPACK to construct numerical solutions and find that these compare favorably with analytical predictions.